Abstract

Interferometric gravitational wave detectors are designed to detect small perturbations in the relative lengths of their kilometer-scale arms that are induced by passing gravitational radiation. An analysis of the effects of imperfect optical alignment on the strain sensitivity of such an interferometer shows that to achieve maximum strain sensitivity at the Laser Interferometer Gravitational Wave Observatory requires that the angular orientations of the optics be within 10-8 rad rms of the optical axis, and the beam must be kept centered on the mirrors within 1 mm. In addition, fluctuations in the input laser beam direction must be less than 1.5 × 10-14 rad/Hz in angle and less than 2.8 × 10-10 m/Hz in transverse displacement for frequencies f > 150 Hz in order that they not produce spurious noise in the gravitational wave readout channel. We show that seismic disturbances limit the use of local reference frames for angular alignment at a level approximately an order of magnitude worse than required. A wave-front sensing scheme that uses the input laser beam as the reference axis is presented that successfully discriminates among all angular degrees of freedom and permits the implementation of a closed-loop servo control to suppress the environmentally driven angular fluctuations sufficiently.

© 1998 Optical Society of America

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  1. A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Science 256, 325–333 (1992).
    [CrossRef] [PubMed]
  2. A. Giazotto, “The VIRGO experiment: status of the art,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, F. Ronga, eds. (World Scientific, Singapore, 1995), pp. 86–99.
  3. K. Danzmann, “GEO 600—600-m laser interferometric gravitational wave antenna,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, F. Ronga, eds. (World Scientific, Singapore, 1995), pp. 100–111.
  4. K. Tsubono, “300-m laser interferometer gravitational wave detector (TAMA300) in Japan,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, F. Ronga, eds. (World Scientific, Singapore, 1995), pp. 112–114.
  5. M. E. Gertsenshtein, V. I. Pustovoit, “On the detection of low frequency gravitational waves,” Sov. Phys. JETP 16, 433–435 (1963); R. Weiss, “Electromagnetically coupled broadband gravitational antenna,” Mass. Inst. Technol. Res. Lab. Electron. Q. Rep. 105, 54–76 (1972).
  6. R. Drever, J. Hough, W. Edelstein, J. Pugh, W. Martin, “A gravity-wave detector using an optical resonator,” in Proceedings of the Ninth International Conference on General Relativity and Gravitation, E. Schmutzer, ed. (VEB, Berlin, 1980), pp. 265–267.
  7. R. W. P. Drever and colleagues, “Gravitational wave detectors using laser interferometers and optical cavities: ideas, principles and prospects,” in Quantum Optics, Experimental Gravity and Measurement Theory, P. Meystre, M. O. Scully, eds. (PlenumNew York, 1983), pp. 503–524; H. Billing, K. Maischberger, A. Ruediger, R. Schilling, L. Schnupp, W. Winkler, “The Munich gravitational wave detector using laser interferometry,” ibid., pp. 525–566.
  8. M. W. Regehr, F. J. Raab, S. E. Whitcomb, “Demonstration of a power-recycled Michelson interferometer with Fabry–Perot arms by frontal modulation,” Appl. Opt. 20, 1507–1509 (1995).
  9. A. Schenzle, R. DeVoe, G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982);R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
    [CrossRef]
  10. L. Schnupp, Max Planck Institute for Quantum Optics, Garching, Germany (personal communication, 1986).
  11. Y. Hefetz, N. Mavalvala, D. Sigg, “Principles of calculating alignment signals in complex optical interferometers,” J. Opt. Soc. Am. B 107, 1597–1605 (1997).
    [CrossRef]
  12. L. Schnupp, Max Planck Institute for Quantum Optics, Garching, Germany (personal communication, 1989); T. M. Niebauer, R. Schilling, K. Danzmann, A. Rüdiger, W. Winkler, “Nonstationary shot noise and its effect on the sensitivity of interferometers,” Phys. Rev. A 43, 5022–5029 (1991).
  13. The eigenvalue of this mode (u2) depends somewhat on the quality of the carrier interference at the antisymmetric port because any noninterfering carrier power contributes to the shot-noise level and thus will tend to increase the eigenvalue. However, in these calculations the shot-noise-producing power is always dominated by the sideband power, and for realistic estimates of the carrier contribution the u2 eigenvalue does not increase by more than 50%. The sideband power dominates because, even though the carrier interference is perfect when there are no misalignments, to better simulate the real interferometer the chosen modulation index is optimal for the expected level of imperfect carrier interference, where of the order of 10-3 of the carrier power in the recycling cavity leaks from the antisymmetric port.
  14. J. A. Arnaud, “Degenerate optical cavities,” Appl. Opt. 8, 189–195 (1969).
    [CrossRef] [PubMed]
  15. See G. González, P. Saulson, “Brownian-motion of a mass suspended by an anelastic wire,” J. Acoust. Soc. Am. 96, 207–212 (1994); A. Gillespie, F. Raab, “Thermally excited vibrations of the mirrors of laser interferometer gravitational-wave detectors,” Phys. Rev. D. 52, 577–585 (1995) for thermal noise applied to translational degrees of freedom. The theory can be extended to rotational degrees of freedom.
  16. S. Kawamura, M. Zucker, “Mirror-orientation noise in a Fabry–Perot interferometer gravitational wave detector,” Appl. Opt. 33, 3912–3918 (1994).
    [CrossRef] [PubMed]
  17. E. Morrison, B. J. Meers, D. I. Robertson, H. Ward, “Experimental demonstration of an automatic alignment system for optical interferometers,” Appl. Opt. 33, 5037–5040 (1994);“Automatic alignment of optical interferometers,” Appl. Opt. 33, 5041–5049 (1994).
    [CrossRef] [PubMed]
  18. J. Giaime, P. Saha, D. Shoemaker, L. Sievers, “A passive vibration isolation stack for LIGO: design, modeling, and testing,” Rev. Sci. Instrum. 67, 208–214 (1996).
    [CrossRef]
  19. A. Rohay, “Ambient ground vibration measurements at the Livingston, Louisiana LIGO Site,” internal report, LIGO-C961022-A-D (LIGO Document Control Center, California Institute of Technology, 1996).
  20. T. Thompson, W. Miller, E. Ponslet, “LIGO seismic isolation system preliminary design review document,” internal report, LIGO-C970251-00-D (LIGO Document Control Center, California Institute of Technology, 1997).
  21. N. Mavalvala, D. Sigg, D. Shoemaker, “Experimental test of an alignment sensing scheme for a gravitational-wave interferometer,” (to be published).

1997

Y. Hefetz, N. Mavalvala, D. Sigg, “Principles of calculating alignment signals in complex optical interferometers,” J. Opt. Soc. Am. B 107, 1597–1605 (1997).
[CrossRef]

1996

J. Giaime, P. Saha, D. Shoemaker, L. Sievers, “A passive vibration isolation stack for LIGO: design, modeling, and testing,” Rev. Sci. Instrum. 67, 208–214 (1996).
[CrossRef]

1995

M. W. Regehr, F. J. Raab, S. E. Whitcomb, “Demonstration of a power-recycled Michelson interferometer with Fabry–Perot arms by frontal modulation,” Appl. Opt. 20, 1507–1509 (1995).

1994

See G. González, P. Saulson, “Brownian-motion of a mass suspended by an anelastic wire,” J. Acoust. Soc. Am. 96, 207–212 (1994); A. Gillespie, F. Raab, “Thermally excited vibrations of the mirrors of laser interferometer gravitational-wave detectors,” Phys. Rev. D. 52, 577–585 (1995) for thermal noise applied to translational degrees of freedom. The theory can be extended to rotational degrees of freedom.

S. Kawamura, M. Zucker, “Mirror-orientation noise in a Fabry–Perot interferometer gravitational wave detector,” Appl. Opt. 33, 3912–3918 (1994).
[CrossRef] [PubMed]

E. Morrison, B. J. Meers, D. I. Robertson, H. Ward, “Experimental demonstration of an automatic alignment system for optical interferometers,” Appl. Opt. 33, 5037–5040 (1994);“Automatic alignment of optical interferometers,” Appl. Opt. 33, 5041–5049 (1994).
[CrossRef] [PubMed]

1992

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

1982

A. Schenzle, R. DeVoe, G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982);R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

1969

1963

M. E. Gertsenshtein, V. I. Pustovoit, “On the detection of low frequency gravitational waves,” Sov. Phys. JETP 16, 433–435 (1963); R. Weiss, “Electromagnetically coupled broadband gravitational antenna,” Mass. Inst. Technol. Res. Lab. Electron. Q. Rep. 105, 54–76 (1972).

Abramovici, A.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Althouse, W. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Arnaud, J. A.

Brewer, G.

A. Schenzle, R. DeVoe, G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982);R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Danzmann, K.

K. Danzmann, “GEO 600—600-m laser interferometric gravitational wave antenna,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, F. Ronga, eds. (World Scientific, Singapore, 1995), pp. 100–111.

DeVoe, R.

A. Schenzle, R. DeVoe, G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982);R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Drever, R.

R. Drever, J. Hough, W. Edelstein, J. Pugh, W. Martin, “A gravity-wave detector using an optical resonator,” in Proceedings of the Ninth International Conference on General Relativity and Gravitation, E. Schmutzer, ed. (VEB, Berlin, 1980), pp. 265–267.

Drever, R. W. P.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

R. W. P. Drever and colleagues, “Gravitational wave detectors using laser interferometers and optical cavities: ideas, principles and prospects,” in Quantum Optics, Experimental Gravity and Measurement Theory, P. Meystre, M. O. Scully, eds. (PlenumNew York, 1983), pp. 503–524; H. Billing, K. Maischberger, A. Ruediger, R. Schilling, L. Schnupp, W. Winkler, “The Munich gravitational wave detector using laser interferometry,” ibid., pp. 525–566.

Edelstein, W.

R. Drever, J. Hough, W. Edelstein, J. Pugh, W. Martin, “A gravity-wave detector using an optical resonator,” in Proceedings of the Ninth International Conference on General Relativity and Gravitation, E. Schmutzer, ed. (VEB, Berlin, 1980), pp. 265–267.

Gertsenshtein, M. E.

M. E. Gertsenshtein, V. I. Pustovoit, “On the detection of low frequency gravitational waves,” Sov. Phys. JETP 16, 433–435 (1963); R. Weiss, “Electromagnetically coupled broadband gravitational antenna,” Mass. Inst. Technol. Res. Lab. Electron. Q. Rep. 105, 54–76 (1972).

Giaime, J.

J. Giaime, P. Saha, D. Shoemaker, L. Sievers, “A passive vibration isolation stack for LIGO: design, modeling, and testing,” Rev. Sci. Instrum. 67, 208–214 (1996).
[CrossRef]

Giazotto, A.

A. Giazotto, “The VIRGO experiment: status of the art,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, F. Ronga, eds. (World Scientific, Singapore, 1995), pp. 86–99.

González, G.

See G. González, P. Saulson, “Brownian-motion of a mass suspended by an anelastic wire,” J. Acoust. Soc. Am. 96, 207–212 (1994); A. Gillespie, F. Raab, “Thermally excited vibrations of the mirrors of laser interferometer gravitational-wave detectors,” Phys. Rev. D. 52, 577–585 (1995) for thermal noise applied to translational degrees of freedom. The theory can be extended to rotational degrees of freedom.

Gürsel, Y.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Hefetz, Y.

Y. Hefetz, N. Mavalvala, D. Sigg, “Principles of calculating alignment signals in complex optical interferometers,” J. Opt. Soc. Am. B 107, 1597–1605 (1997).
[CrossRef]

Hough, J.

R. Drever, J. Hough, W. Edelstein, J. Pugh, W. Martin, “A gravity-wave detector using an optical resonator,” in Proceedings of the Ninth International Conference on General Relativity and Gravitation, E. Schmutzer, ed. (VEB, Berlin, 1980), pp. 265–267.

Kawamura, S.

S. Kawamura, M. Zucker, “Mirror-orientation noise in a Fabry–Perot interferometer gravitational wave detector,” Appl. Opt. 33, 3912–3918 (1994).
[CrossRef] [PubMed]

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Martin, W.

R. Drever, J. Hough, W. Edelstein, J. Pugh, W. Martin, “A gravity-wave detector using an optical resonator,” in Proceedings of the Ninth International Conference on General Relativity and Gravitation, E. Schmutzer, ed. (VEB, Berlin, 1980), pp. 265–267.

Mavalvala, N.

Y. Hefetz, N. Mavalvala, D. Sigg, “Principles of calculating alignment signals in complex optical interferometers,” J. Opt. Soc. Am. B 107, 1597–1605 (1997).
[CrossRef]

N. Mavalvala, D. Sigg, D. Shoemaker, “Experimental test of an alignment sensing scheme for a gravitational-wave interferometer,” (to be published).

Meers, B. J.

Miller, W.

T. Thompson, W. Miller, E. Ponslet, “LIGO seismic isolation system preliminary design review document,” internal report, LIGO-C970251-00-D (LIGO Document Control Center, California Institute of Technology, 1997).

Morrison, E.

Ponslet, E.

T. Thompson, W. Miller, E. Ponslet, “LIGO seismic isolation system preliminary design review document,” internal report, LIGO-C970251-00-D (LIGO Document Control Center, California Institute of Technology, 1997).

Pugh, J.

R. Drever, J. Hough, W. Edelstein, J. Pugh, W. Martin, “A gravity-wave detector using an optical resonator,” in Proceedings of the Ninth International Conference on General Relativity and Gravitation, E. Schmutzer, ed. (VEB, Berlin, 1980), pp. 265–267.

Pustovoit, V. I.

M. E. Gertsenshtein, V. I. Pustovoit, “On the detection of low frequency gravitational waves,” Sov. Phys. JETP 16, 433–435 (1963); R. Weiss, “Electromagnetically coupled broadband gravitational antenna,” Mass. Inst. Technol. Res. Lab. Electron. Q. Rep. 105, 54–76 (1972).

Raab, F. J.

M. W. Regehr, F. J. Raab, S. E. Whitcomb, “Demonstration of a power-recycled Michelson interferometer with Fabry–Perot arms by frontal modulation,” Appl. Opt. 20, 1507–1509 (1995).

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Regehr, M. W.

M. W. Regehr, F. J. Raab, S. E. Whitcomb, “Demonstration of a power-recycled Michelson interferometer with Fabry–Perot arms by frontal modulation,” Appl. Opt. 20, 1507–1509 (1995).

Robertson, D. I.

Rohay, A.

A. Rohay, “Ambient ground vibration measurements at the Livingston, Louisiana LIGO Site,” internal report, LIGO-C961022-A-D (LIGO Document Control Center, California Institute of Technology, 1996).

Saha, P.

J. Giaime, P. Saha, D. Shoemaker, L. Sievers, “A passive vibration isolation stack for LIGO: design, modeling, and testing,” Rev. Sci. Instrum. 67, 208–214 (1996).
[CrossRef]

Saulson, P.

See G. González, P. Saulson, “Brownian-motion of a mass suspended by an anelastic wire,” J. Acoust. Soc. Am. 96, 207–212 (1994); A. Gillespie, F. Raab, “Thermally excited vibrations of the mirrors of laser interferometer gravitational-wave detectors,” Phys. Rev. D. 52, 577–585 (1995) for thermal noise applied to translational degrees of freedom. The theory can be extended to rotational degrees of freedom.

Schenzle, A.

A. Schenzle, R. DeVoe, G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982);R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Schnupp, L.

L. Schnupp, Max Planck Institute for Quantum Optics, Garching, Germany (personal communication, 1986).

L. Schnupp, Max Planck Institute for Quantum Optics, Garching, Germany (personal communication, 1989); T. M. Niebauer, R. Schilling, K. Danzmann, A. Rüdiger, W. Winkler, “Nonstationary shot noise and its effect on the sensitivity of interferometers,” Phys. Rev. A 43, 5022–5029 (1991).

Shoemaker, D.

J. Giaime, P. Saha, D. Shoemaker, L. Sievers, “A passive vibration isolation stack for LIGO: design, modeling, and testing,” Rev. Sci. Instrum. 67, 208–214 (1996).
[CrossRef]

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

N. Mavalvala, D. Sigg, D. Shoemaker, “Experimental test of an alignment sensing scheme for a gravitational-wave interferometer,” (to be published).

Sievers, L.

J. Giaime, P. Saha, D. Shoemaker, L. Sievers, “A passive vibration isolation stack for LIGO: design, modeling, and testing,” Rev. Sci. Instrum. 67, 208–214 (1996).
[CrossRef]

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Sigg, D.

Y. Hefetz, N. Mavalvala, D. Sigg, “Principles of calculating alignment signals in complex optical interferometers,” J. Opt. Soc. Am. B 107, 1597–1605 (1997).
[CrossRef]

N. Mavalvala, D. Sigg, D. Shoemaker, “Experimental test of an alignment sensing scheme for a gravitational-wave interferometer,” (to be published).

Spero, R. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Thompson, T.

T. Thompson, W. Miller, E. Ponslet, “LIGO seismic isolation system preliminary design review document,” internal report, LIGO-C970251-00-D (LIGO Document Control Center, California Institute of Technology, 1997).

Thorne, K. S.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Tsubono, K.

K. Tsubono, “300-m laser interferometer gravitational wave detector (TAMA300) in Japan,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, F. Ronga, eds. (World Scientific, Singapore, 1995), pp. 112–114.

Vogt, R. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Ward, H.

Weiss, R.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Whitcomb, S. E.

M. W. Regehr, F. J. Raab, S. E. Whitcomb, “Demonstration of a power-recycled Michelson interferometer with Fabry–Perot arms by frontal modulation,” Appl. Opt. 20, 1507–1509 (1995).

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Zucker, M.

Zucker, M. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Appl. Opt.

J. Acoust. Soc. Am.

See G. González, P. Saulson, “Brownian-motion of a mass suspended by an anelastic wire,” J. Acoust. Soc. Am. 96, 207–212 (1994); A. Gillespie, F. Raab, “Thermally excited vibrations of the mirrors of laser interferometer gravitational-wave detectors,” Phys. Rev. D. 52, 577–585 (1995) for thermal noise applied to translational degrees of freedom. The theory can be extended to rotational degrees of freedom.

J. Opt. Soc. Am. B

Y. Hefetz, N. Mavalvala, D. Sigg, “Principles of calculating alignment signals in complex optical interferometers,” J. Opt. Soc. Am. B 107, 1597–1605 (1997).
[CrossRef]

Phys. Rev. A

A. Schenzle, R. DeVoe, G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982);R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Rev. Sci. Instrum.

J. Giaime, P. Saha, D. Shoemaker, L. Sievers, “A passive vibration isolation stack for LIGO: design, modeling, and testing,” Rev. Sci. Instrum. 67, 208–214 (1996).
[CrossRef]

Science

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Sov. Phys. JETP

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Other

R. Drever, J. Hough, W. Edelstein, J. Pugh, W. Martin, “A gravity-wave detector using an optical resonator,” in Proceedings of the Ninth International Conference on General Relativity and Gravitation, E. Schmutzer, ed. (VEB, Berlin, 1980), pp. 265–267.

R. W. P. Drever and colleagues, “Gravitational wave detectors using laser interferometers and optical cavities: ideas, principles and prospects,” in Quantum Optics, Experimental Gravity and Measurement Theory, P. Meystre, M. O. Scully, eds. (PlenumNew York, 1983), pp. 503–524; H. Billing, K. Maischberger, A. Ruediger, R. Schilling, L. Schnupp, W. Winkler, “The Munich gravitational wave detector using laser interferometry,” ibid., pp. 525–566.

L. Schnupp, Max Planck Institute for Quantum Optics, Garching, Germany (personal communication, 1986).

A. Giazotto, “The VIRGO experiment: status of the art,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, F. Ronga, eds. (World Scientific, Singapore, 1995), pp. 86–99.

K. Danzmann, “GEO 600—600-m laser interferometric gravitational wave antenna,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, F. Ronga, eds. (World Scientific, Singapore, 1995), pp. 100–111.

K. Tsubono, “300-m laser interferometer gravitational wave detector (TAMA300) in Japan,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizella, F. Ronga, eds. (World Scientific, Singapore, 1995), pp. 112–114.

A. Rohay, “Ambient ground vibration measurements at the Livingston, Louisiana LIGO Site,” internal report, LIGO-C961022-A-D (LIGO Document Control Center, California Institute of Technology, 1996).

T. Thompson, W. Miller, E. Ponslet, “LIGO seismic isolation system preliminary design review document,” internal report, LIGO-C970251-00-D (LIGO Document Control Center, California Institute of Technology, 1997).

N. Mavalvala, D. Sigg, D. Shoemaker, “Experimental test of an alignment sensing scheme for a gravitational-wave interferometer,” (to be published).

L. Schnupp, Max Planck Institute for Quantum Optics, Garching, Germany (personal communication, 1989); T. M. Niebauer, R. Schilling, K. Danzmann, A. Rüdiger, W. Winkler, “Nonstationary shot noise and its effect on the sensitivity of interferometers,” Phys. Rev. A 43, 5022–5029 (1991).

The eigenvalue of this mode (u2) depends somewhat on the quality of the carrier interference at the antisymmetric port because any noninterfering carrier power contributes to the shot-noise level and thus will tend to increase the eigenvalue. However, in these calculations the shot-noise-producing power is always dominated by the sideband power, and for realistic estimates of the carrier contribution the u2 eigenvalue does not increase by more than 50%. The sideband power dominates because, even though the carrier interference is perfect when there are no misalignments, to better simulate the real interferometer the chosen modulation index is optimal for the expected level of imperfect carrier interference, where of the order of 10-3 of the carrier power in the recycling cavity leaks from the antisymmetric port.

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Figures (6)

Fig. 1
Fig. 1

Schematic optical layout of a power recycled interferometric gravitational wave detector. The input light is phase modulated with a rf waveform to generate error signals for the length and alignment degrees of freedom. The input test masses (ITM1 and ITM2) and the recycling mirror (RM) are partially transmitting mirrors, whereas the end test masses (ETM1 and ETM2) have maximum reflectivity; the beam splitter (BS) splits the light 50–50 between the two arms. The gravitational wave signal is derived from the light at the antisymmetric port; the recycling cavity port represents a small fraction of the light in the recycling cavity, which is typically provided by the antireflection-coated surface of the beam splitter or an ITM. Also indicated is the sign convention for the alignment angles. The symbol next to each optic shows the rotation axes (vectors) for horizontal and vertical tilts. A positive tilt angle corresponds to a right-handed rotation about the axis defined by the rotation vector.

Fig. 2
Fig. 2

Contour of reduction of power coupling with misalignment for a half-symmetric resonator. The ellipse’s shape and orientation correspond to a cavity with g = 1/3 (at some arbitrary power reduction factor), for which α = π/8 and (σ12)2 = 5.83.

Fig. 3
Fig. 3

Pictorial representation of the two most sensitive alignment degrees of freedom for the interferometer’s shot-noise-limited signal-to-noise ratio. The filled bars represent the mirrors in the perfectly aligned interferometer, and the open bars indicate the (exaggerated) angles in the labeled mode.

Fig. 4
Fig. 4

Schematic of a LIGO test mass and its support, isolation, and suspension systems. The dominant angular motion of the mirror is expected to be in pitch, θ M , and is induced primarily by translation Z G of the facility foundation F, causing translation of the seismic isolation stack S and thus of the pendulum suspension point. The wire sling suspension P converts this translation into a torque (inset). The pivot point CP where the wire is joined to the test mass lies a small distance h above its center of gravity CG. (h is chosen to make the mirror gravitationally stable with an appropriate pitch eigenfrequency.) Acceleration of the suspension support (S) thus induces a torque about the mirror’s pitch axis.

Fig. 5
Fig. 5

Block diagram of electronic compensation for an equivalent single-input–single-output loop design to control one composite angular degree of freedom (for example, u 1). The control compensation is tailored to counteract the microseismic motion (which peaks at 0.15 Hz) and the expected resonant amplification of the seismic isolation stacks (at their eigenfrequency of 1.4 Hz) by inclusion of two tuned resonant gain sections. Sharp low-pass filtering above the 5.2-Hz unity gain frequency is provided to prevent infiltration of residual sensing (e.g., photocurrent shot) and electronic noise in the LIGO measurement band, above 40 Hz. BP’s, bandpass filters; LPF, low-pass filter; WFS, wave-front sensor.

Fig. 6
Fig. 6

Top, spectral density of ambient horizontal seismic vibration measured at the Livingston, Louisiana, LIGO site (gray curve; this is the Z G input in Fig. 4), and the predicted vibration spectrum at the pendulum support point (black curve; this is the Z S input in Fig. 4). Bottom, predicted open-loop fluctuations of the mirror pitch angle (gray curve; the modes at 0.5 and 0.74 Hz are unresolved in this plot), and the residual pitch angle fluctuations when the mirror is controlled by the servo described in Fig. 5 (black curve). The integral of this spectrum equals 8 × 10-9 rad rms.

Tables (4)

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Table 1 Eigenvalues and Eigenvectors of the Five-Dimensional Misalignment Variance Ellipsoid for the Shot-Noise-Limited Signal-to-Noise Ratio of the Interferometera

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Table 2 Alignment Sensing Parameters Defined in Eq. (16) for Each Sensing Porta

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Table 3 The Five Wave-Front Sensor Signals for a Specific Choice of Sensor Positions, Gouy Phases, and RF Phasesa

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Table 4 Wave-Front Sensor Angle and Angle-Equivalent Shot Noise Sensitivitiesa

Equations (20)

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Δ θ ETM Δ θ ITM θ ETM ¯ θ ITM ¯ RM 1 2 0 - 1 0 1 0 - 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 2 θ ITM 1 θ ETM 1 θ ITM 2 θ ETM 2 θ RM .
S sens θ = S sens 0 1 - ½ θ H θ .
H ij = d 2 d θ i d θ j   S sens θ
ε = 2   i = 1 5 ψ i σ i 2 ,
P θ 1 ,   θ 2 / P max exp - R 2 g θ 1 + θ 2 2 / ω 0 2 × exp - LRg θ 1 / ω 0 2 1 - R ω 0 2 g θ 1 2 + 2 g θ 1 θ 2 + θ 2 2 ,
σ 1 2 / σ 2 2 = 1 + g + 5 g 2 - 2 g + 1 1 / 2 1 + g - 5 g 2 - 2 g + 1 1 / 2 .
N P c + ³ / P sb 1 / 2 ,
η g , rc n ,   m = n + m + 1 tan - 1 l rc / z 0 ,
Φ rt = Φ θ = 0 + ½ θ H θ ,
δ Φ rt t = ½ Δ θ H Δ θ + Δ θ H θ t + ½ θ t H θ t .
δ Φ bs f = 8 × 10 - 11 rad Hz i = 1 2 - 1 i α i , rms 10 - 8   rad × α i f 7 × 10 - 16   rad / Hz ,
E 0 x ,   z ,   t = 1 - x ib 2 2 - α ib 2 2 U 0 x ,   z + x ib t 1 + i z / z 0 - i α ib t U 1 x ,   z e i ω t ,
δ Φ bs t = 2.6 × 10 - 11 0.91 Δ θ ETM + 0.41 Δ θ ITM 10 - 8   rad α ib t 10 - 8 + 0.19 0.90 Δ θ ETM + 0.43 Δ θ ITM 10 - 8   rad x ib t 10 - 8 rad .
x ib ,   y ib < 8.1 × 10 - 9 / Hz , α ib ,   β ib < 1.5 × 10 - 9 / Hz
x ib t = x 0 cos ω a t ,     α ib t = α 0 cos ω a t .
WFS η ,   Θ ,   Γ = P in f Γ f split k PD × i = 1 5   A i Θ i   cos η - η i cos ω m t - ϕ Di ,
| S WFS | P in f Γ f split k PD A WFS / θ D ,
N j 2 e η P j / hv 0 1 / 2 ,
α ˜ WFS = N j hv 0 / η | S WFS | .
M t     U 0 = 1 - 4 α 2 cos 2 ω a t U 0 + 2 i α   cos   ω a tU 1 ,

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